8C Pi TEACHER INSTRUCTIONS Unit 8 – Circles Level 1: I can use relationships between angles and arcs in circles to solve for missing measures. G-‐C 2 Level 2: I can use relationships between secants, chords, and tangents in circles to solve for missing measures. G-‐C 2, G-‐C 4 Level 3: I can use similarity to calculate arc length and area of a sector. G-‐C 5 Level 4: I can prove relationships between secants, chords, and tangent in circles. G-‐C 2 Circles: G-‐C Understand and apply theorems about circles 2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. 4. (+) Construct a tangent line from a point outside a given circle to the circle. Find arc lengths and areas of sectors of circles 5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Materials: TI-‐Nspire Navigator System TI-‐Nspire Handhelds Student Handout – Pi, one cut out circle for each student (included in this document) Files: Bell 8E.tns, HW Quiz 8E.tns, Exit 8E.tns, Pi.tns, Pi.flipchart, Pi_QP.tns Teacher Notes: Exploring_Diameter_and_Circumference_Teacher.pdf Resource: Geometry Nspired – Exploring_Diameter_and_Circumference 1. Bell 8C 2. Math Content: What is pi? Circumference and area of a circle, Area of shaded regions with circles, probability 3. Exit Ticket Pi 1. What is π? 2. How would you describe the general shape of the data? 3. What function represents the relationship between the Circumference & Diameter of the circles? 4. What is π? 5. How would you describe the general shape of the data? 6. What function represents the relationship between the Circumference & Diameter of the circles? Name______________________________________ 7. A bicycle mechanic wants to put a strip of plastic between the tube and tire of a 26-‐in. diameter bicycle tire. To the nearest inch, how long should the strip of plastic be? 8. The most common tire size circumference is 67 inches. To the nearest tenth, what is the length of the radius of the tire? 9. The diameter of a basketball rim is 18 in. A standard basketball has a circumference of 30 in. About how much room is there between the ball and the rim in a shot in which the ball goes exactly in the center of the rim? 10. Three tennis balls are packaged in a pressurized can, one on top of the other. Is the height of the can or its circumference greater? Justify your answer. 11. A circle is inscribed in a square with a side of 12 ft. What is 12. A square with a side of 12 ft is inscribed in a circle. What is the area of the region between the square and the circle? the area of the region between the square and the circle? 13. A square with a diagonal of 10 ft is inscribed in a circle. What 14. A circle is inscribed in a square with a diagonal of 12 ft. What is the area of the region between the square and the circle? is the area of the region between the square and the circle? 15. What is the area of the shaded region? 16. What is the area of the shaded region? 17. What is the area of the shaded region? 19. What is the area of the shaded region? 18. What is the area of the shaded region? 20. Triangle ABC is inscribed in circle O. Angle C is a right angle, and AB=6. What is the area of the region inside the circle but outside the triangle? Draw and label a diagram and show all of your work. 21. You win if you throw a ball in the part of the board that is shaded blue. Should you play the game? 22. In the figure, a square is inscribed in a circle with diameter d. What is the sum of the areas of the shaded regions, in terms of d? ⎛ π 1⎞ ⎛ π 1⎞ A. d 2 ⎜ − ⎟ B. d 2 ⎜ − ⎟ ⎝ 4 2⎠ ⎝ 4 4⎠ ! ! ⎛ π 1⎞ C. d 2 ⎜ − ⎟ D. d 2 π − 2 ! ⎝ 2 2⎠ ! E. d 2 π −1 ! 23. The target is 10 feet in diameter. The inside 50-‐point circle is 2 feet in diameter. Each ring is 1 foot wide. Point values for each ring are listed. ( ( ) ) a. What is the area of the bull’s-‐eye? b. What proportion of the area of the target is the bull’s-‐eye? c. What is the area of the 10-‐point ring? d. What is the total area of all of the shaded rings (including the bull’s-‐eye)? e. What proportion of the area of the target is shaded? f. What proportion of the area of the target is not shaded? g. Is an archer more likely to score 40 points with two arrows by scoring 10 first and 30 second or by scoring 20 points with both arrows? Explain your reasoning.
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